The sum of the first and third term of an arithmetic progression is $12$ and the product of the first and second term is $24$. Find the first term.

If $S_n = 1 + \frac{1}{2} + \frac{1}{2^2} + \dots + \frac{1}{2^{n-1}}$,then the least integral value of $n$ such that $2 - S_n < \frac{1}{100}$ is

The sum of the series $1^2 \cdot 2 + 2^2 \cdot 3 + 3^2 \cdot 4 + \dots$ to $n$ terms is:

The fifth term of the $H.P.$,$2, 2\frac{1}{2}, 3\frac{1}{3}, \dots$ will be

If ${S_k}$ denotes the sum of the first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively,then ${S_{kn}}/{S_n}$ is independent of $n$ if:

The $H.M.$ between the roots of the equation $x^2 - 10x + 11 = 0$ is